Seeking a precise, abstract definition of recursive decomposability, we introduce spined categories. These are categories equipped with sufficient additional structure to admit a well-behaved notion of recursive decomposition. In our main result (Theorem REF ) we provide a uniform construction of an invariant called the triangulation functor which is defined on (objects of) spined categories. The triangulation functor coincides with tree-width in the case of graphs. In particular, we obtain a characterization of our abstract analogue of tree-width as a functor satisfying certain maximality properties (Theorem REF ). This characterization is closely related (but not identical) to Halin's [1]} definition of tree-width as the maximal graph parameter sharing certain properties with the Hadwiger number and chromatic number. We show that the triangulation functors of other spined categories encompass several tree-width-like invariants such as hypergraph tree-width (Theorem REF ), and the tree-width of the modular quotient in the category of modular partition functions (Example REF ).
